Geometry and Exact Soliton Solutions of the Integrable su(3)-

Transcription

1 Geometry and Exact Soliton Solutions of the Integrable su(3)-spin Systems Eurasian national university, Astana, Kazakhstan Varna, Bulgaria 2018

2 Introduction The vector nonlinear Schrödinger equation is associated with symmetric space SU(n + 1)/S(U(1) U(n)). The special case n = 2 of such symmetric space is associated with the Manakov system iq 1t + q 1xx + 2( q q 2 2 )q 1 = 0, (1) iq 2t + q 2xx + 2( q q 2 2 )q 2 = 0. (2) Geometrical equivalent counterpart of the Manakov system is the coupled spin systems A t + A A xx + u 1 A x + 2v 1 H A = 0, (3) B t + B B xx + u 2 B x + 2v 2 H B = 0, (4) where spin vectors A = (A 1, A 2, A 3 ) and B = (B 1, B 2, B 3 ), A 2 = B 2 = 1, H = (0, 0, 1) T is the constant magnetic field, u j and v j are coupling potentials.

3 In matrix form the coupled spin systems (3)-(4) ia t [A, A xx] + iu 1 A x + v 1 [σ 3, A] = 0, (5) ib t [B, B xx] + iu 2 B x + v 2 [σ 3, B] = 0, (6) where ( A3 A A = ) A +, A 3 A 2 = I = diag(1, 1), A ± = A 1 ± ia 2. (7) ( B3 B B = ) B +, B 2 = I = diag(1, 1), B ± = B B 1 ± ib 2. (8) 3

4 Lax representation of the coupled spin equation hierarchy We present two possible versions of the Lax representation (LR) for the coupled spin equation hierarchy. LR type - I The first type of LR for the coupled spin equation hierarchy reads as Y x = iλpy, (9) Y t = where λ is a spectral parameter and P = K N λ j V j Y, (10) j=1 2A 3 K 2A 2(1+A 3 )B 1+B 3 2A + (2A 3 + K ) 2A + B 1+B 3 2(1+A 3 )B + 1+B 3 2A B + 1+B 3 K 2, (11)

5 with K = (1 + A 3 )(1 B 3 )(1 + B 3 ) 1. The compatibility condition of this system gives the coupled spin equation hierarchy. As the particular example, let us consider the case when N = 2. Then the set of equations (9)-(10) takes the form where Y x = iλpy, (12) Y t = ( λ 2 V 2 + λv 1 ) Y, (13) V 2 = 2iP, V 1 = PP x. (14) The compatibility condition of the equations (12)-(13) gives the coupled spin equation (5)-(6).

6 LR type - II The second type of LR for the coupled spin equation hierarchy can be written in the following form Here Y x = iλqy, (15) Y t = N λ j W j Y. (16) j=1 Q = Q 1 + Q 2, (17)

7 where Q 1 = Q 2 = 0 A 1 A 2 A 3 A 1 0 A 3 A 2 A 2 A 3 0 A 1 A 3 A 2 A B 1 B 2 B 3 B 1 0 B 3 B 2 B 2 B 3 0 B 1 B 3 B 2 B 1 0, (18). (19) From the compatibility condition of the set of equations (15)-(16) Y xt = Y tx we obtain the coupled spin equation hierarchy.

8 Relation between solutions of the coupled spin equation and the Manakov system Let A j and B j be the solution of the coupled spin equation (5)-(6). Then the solution of the Manakov system (1)-(2) is given by q 1 = q 2 = Re 2iν W (1 + A 3 ), (20) Ze 2iν W (1 + A 3 ). (21)

9 Here W = 2 + (1 + A 3)(1 B 3 ) 1 + B 3 = 2 + K, R = WA x MA, Z = W [(1 + A 3 )(1 + B 3 ) 1 B ] x M[(1 + A 3 )(1 + B 3 ) 1 B ]. M = A 3x + A+ A x 1 + A 3 + A 3x (1 B 3 ) 1 + B 3 + (1 + A 3)B + B x (1 + B 3 ) 2 (1 + A 3)(1 B 3 )B 3x (1 + B 3 ) 2, [ ν = x 1 A1 A 2x A 1x A 2 (1 + A 3 )W (1 + A 3)(B 1x B 2 B 1 B 2x ) (1 + B 3 ) 2 W ].

10 Gauge equivalence between the Γ-spin system and the Manakov system In this section, we want to present the gauge equivalent counterpart of the Manakov system when n = 3. The Lax representation of the Manakov system (1)-(2) has the form Here Φ x = UΦ, (22) Φ t = V Φ. (23) U = iλσ + U 0, V = 2iλ 2 Σ + 2λU 0 + V 0 (24)

11 with Σ = V 0 = i, U 0 = 0 q 1 q 2 q q q q 2 2 q 1x q 2x q 1x q 1 2 q 1 q 2 q 2x q 2 q 1 q 2 2 Let us now consider the gauge transformation Then Ψ obeys the equations, (25). (26) Ψ = g 1 Φ, g = Φ λ=0. (27) Ψ x = U Ψ, (28) Ψ t = V Ψ, (29)

12 where U = iλγ, V = 2iλ 2 Γ λ[γ, Γ x]. (30) Here Γ = g 1 Σg, Γ 2 = I (31) and Γ = Γ 11 Γ 12 Γ 13 Γ 21 Γ 22 Γ 23 Γ 31 Γ 32 Γ 33 su(3). (32)

13 Elements of the Γ matrix satisfy some restrictions Γ 33 = (1 + Γ 11 + Γ 22 ), Γ ij = Γ ji, (33) and Γ ik Γ kj + Γ i(k+1) Γ (k+1)i + Γ i(k+2) Γ (k+2)i = 0, (i = k = j),(34) Γ ik Γ ki + Γ i(k+1) Γ (k+1)i + Γ i(k+2) Γ (k+2)i = 1. (35) The compatibility condition of the equations (28)-(29) gives iγ t [Γ, Γ xx] = 0. (36)

14 Relation between solutions of the coupled spin equation and the Γ-spin system In the previous sections we have shown that to the one and same set of equations - the Manakov system (1)-(2), correspond two spin systems: the coupled spin equation (5)-(6) and the Γ-spin system (36). It tells us that between these two spin systems there must be some exact relation/correspondence. In other words, the 2-layer spin equation (5)-(6) and the Γ-spin system (36) are equivalent to each other by some exact transformations. Below we will present these transformations.

15 Direct transformation According to the transformation, in terms of the spin vectors A and B, the elements of the Γ-spin system are expressed as where Γ = K 2A 3 K 2A 2(1+A 3 )B 1+B 3 2A + (2A 3 + K ) 2A + B 1+B 3 2(1+A 3 )B + 1+B 3 2A B + 1+B 3 K 2, (37) K = (1 + A 3)(1 B 3 ) 1 + B 3. (38) This is the direct transformation. This transformation allows us to find solutions of the Γ-spin system (36) if we know the solutions of the coupled spin equation (5)-(6).

16 Inverse transformation According to the inverse transformation, solutions of the coupled spin equation can be expressed by the components of the Γ-spin system as ( ) 1 Γ11 Γ A = 22 2Γ 12, (39) 1 Γ 33 2Γ 21 Γ 22 Γ 11 B = 1 ( Γ11 Γ 33 2Γ 13 1 Γ 22 2Γ 31 Γ 33 Γ 11 ). (40) The transformations (39)-(40) is called the inverse transformation. Using the inverse transformation, we can find solutions of the coupled spin equation (5)-(6), if we know the solutions of the Γ-spin system (36).

17 Darboux transformation and exact solutions of the Γ-spin system In this section, we construct the DT for the equation (36). To do this, let us consider the following transformation of solutions of the equations (28)-(29) where Φ = LΨ, (41) L = λn I. (42) We require that Φ satisfies the same Lax representation as (28)-(29) so that where U V depend on Γ as U V on Γ. Φ x = U Φ, (43) Φ t = V Φ, (44)

18 The matrix L obeys the following equations L x + LU = U L, (45) L t + LV = V L. (46) These equations yield the following equations for N N x = iγ iγ, (47) N t = Γ Γ x + ΓΓ x (48) and Γ = NΓN 1. (49) Also we have the following useful second form of the DT for Γ : Γ = Γ in x. (50)

19 Let us consider the following set of equations H x = iγhλ, (51) H t = 2iΓHΛ 2 + ΓΓ x HΛ, (52) where λ Λ = 0 λ 2 0, (53) 0 0 λ 3 det H = 0 and λ k are complex constants. We now assume that the matrix N can be written as: n 11 n 12 n 13 N = HΛ 1 H 1 = n 21 n 22 n 23. (54) n 31 n 32 n 33

20 The inverse matrix we write as N 1 = HΛH 1 = 1 m 11 m 12 m 13 m 21 m 22 m 23 (55) n m 31 m 32 m 33 or N 1 = 1 n 22n 33 n 23 n 32 (n 12 n 33 n 13 n 32 ) n 12 n 23 n 13 n 22 (n 21 n 33 n 23 n 31 ) n 11 n 33 n 31 n 13 (n 11 n 23 n 21 n 13 ) n n 32 n 21 n 31 n 22 (n 11 n 32 n 31 n 12 ) n 11 n 22 n 21 n 12 where n = detn and has the form n = n 11 n 22 n 33 + n 12 n 23 n 31 + n 13 n 32 n 21 n 31 n 22 n 13 n 12 n 21 n 33 n 11 n 23 n 32.

21 From these equations follow that N obeys the equations N x = inγn 1 iγ, (57) N t = ΓΓ x NΓΓ x N 1, (58) which are equivalent to Eqs.(57)-(58) as we expected. The Γ and matrix solutions of the system (28)-(29) obey the condition which follow from the equations Φ = Φ 1, Γ = Γ, (59) Φ x = iλφ Γ, (Φ 1 ) x = iλφ 1 Γ 1. (60)

22 Here denote an Hermitian conjugate. After some calculations we came to the formulas ψ 1 (λ 1 ; t, x, y) ψ 2 (λ 1 ; t, x, y) ψ 3 (λ 1 ; t, x, y) H = ψ 2 (λ 1 ; t, x, y) ψ 1 (λ 1 ; t, x, y) 0, (61) ψ 3 (λ 1 ; t, x, y) 0 ψ 1 (λ 1 ; t, x, y) where λ 2 = λ 3 = λ 1, H 1 = 1 ψ 2 1 ψ 1 ψ 2 ψ 1 ψ 3 ψ 1 ψ 2 ( ψ 1 ψ 2 + ψ 2 2 ) ψ 2 ψ 3, (62) 1 ψ 1 ψ 3 ψ 2 ψ 3 ( ψ ψ 2 2 where = ψ ψ ψ 3 2. (63)

23 So finally for the matrix N we get the following expression N = 1 where ɛ ij = λ 1 i n 11 n 12 n 13 ψ 1 ψ 2 ɛ 12 ψ 2 2 λ 1 + ψ ψ 3 2 λ 2 ψ 2 ψ 3 ɛ 12 ψ 1 ψ 3 ɛ 13 ψ 2 ψ 3 ɛ 13 ψ 3 2 λ 1 + ψ ψ 2 2 λ 3 λ 1 j, n 11 = ψ 1 2 λ 1 + ψ 2 2 λ 2 + ψ 3 2 n 12 = ψ 1 ψ 2 ψ 1 ψ 2 + ψ 2 ψ 3 2 (λ λ 1 λ 2 ψ 1 1 n 13 = ψ 1 ψ 3 ψ 1 ψ 3 + ψ 2 2 ψ 3 (λ λ 1 λ 3 ψ 1 1, (64) λ 3, (65) 3 λ2 1 ), (66) 2 λ3 1 ). (67)

24 Hence we can write the DT in terms of the eigenfunctions of the Lax representations (28)-(29) as Γ [1] = 1 n 11m 11 n 12 m 21 n 13 m 31 n 11 m 12 n 12 m 22 n 13 m 32 n 11 m 13 n 12 m n 21 m 11 n 22 m 21 n 23 m 31 n 21 m 12 n 22 m 22 n 23 m 32 n 21 m 13 n 22 m n n 31 m 11 n 32 m 21 n 33 m 31 n 31 m 12 n 32 m 22 n 33 m 32 n 31 m 13 n 32 m

25 To construct the 1-soliton solution of the Γ-spin system (36), now we consider a seed solution In our case the eigenfunctions are given by Γ [0] = Σ. (69) ψ [0] 1 = e θ+iδ 1, ψ [0] 2 = e θ+iδ 2, ψ [0] 3 = e θ+iδ 3, (70) where δ i are complex constants and Then we get θ = θ 1 + iθ 2 = iλ 1 x 2iλ 2 1t. (71) Γ [1] = Γ [1] 11 Γ [1] 12 Γ [1] 13 Γ [1] 21 Γ [1] 22 Γ [1] 23 Γ [1] 31 Γ [1] 32 Γ [1] 33, (72) where n ij and m ij are given by the equations (65)-(67).

26 The 1-soliton solutions of the coupled spin equation Let us now we present the formulas for the 1-soliton solution of the coupled spin equation (5)-(6). Its seed solution we write as A [0] = σ 3, B [0] = σ 3. (73) To find the 1-soliton solution of the coupled spin equation here we use the inverse M-transformation. The inverse transformation allows us to find solutions of the coupled spin equation (5)-(6), if we know the solutions of the Γ-spin system (36). So the 1-soliton solution of the coupled spin equation has the form

27 A [1] = B [1] = 1 1 Γ [1] Γ [1] 22 ( ( ) Γ [1] 11 Γ[1] 22 2Γ [1] 12 2Γ [1] 21 Γ [1] 22, (74) Γ[1] 11 ) Γ [1] 11 Γ[1] 33 2Γ [1] 13 2Γ [1] 31 Γ [1] 33, (75) Γ[1] 11 where Γ [1] ij are given by the formulas (72) and (68).

28 Conclusion In this paper we have presented the DT for the Γ-spin system which is the integrable su(3)-valued spin system. In particular, we have given the explicit formula for its 1-soliton solution. Then we have shown how construct soliton solutions of the coupled spin equation for the two coupled su(2) - valued spin systems. For this purpose we have used the DT formulas of the Γ-spin system. Also the Lax representation of the coupled spin equation is presented. Using this Lax representation, the gauge equivalence between the coupled spin equation and the Manakov system is established. The results obtained in this paper will be useful in the study of nonlinear dynamics of multilayer magnetic systems. Also they will be useful in differential geometry of curves and surfaces to find the integrable deformations of interacting curves and surfaces.

29 References [1] N. A. Kostov, R. Dandoloff, V. S. Gerdjikov, G. G. Grahovski. The Manakov system as two moving interacting curves, In the Proceedings of the International Workshop Complex structures and vector fields, August 21 26, 2006, Sofia, Bulgaria. Eds.: K. Sekigawa, S. Dimiev. World Scientific (2007). [2] C. Senthilkumar, M. Lakshmanan, B. Grammaticos, A. Ramani, Phys. Lett. A (2006). [3] Myrzakulov R., Rahimov F.K., Serikbaev N.S. On the geometry of stationary Heisenberg ferromagnets. In: Non-linear waves: Classical and Quantum Aspects, Kluwer Academic Publishers, Dordrecht, Netherlands, P (2004). [4] Z.S. Yersultanova, M. Zhassybayeva, K. Yesmakhanova, G. Nugmanova, R. Myrzakulov. International Journal of Geometric Methods in Modern Physics, 13, N1, (2016). [5] Chen Hai, Zhou Zi-Xiang. Darboux Transformation with a Double Spectral Parameter for the Myrzakulov-I Equation. Chin. Phys. Lett., 31, N12, (2014).

30 Thank you for attention!